How can we prove Cauchy-Goursat theorem ?

Cauchy - Goursat theorem says, the integral of an analytic function  f(z) over a closed contour C ,  is zero. Mathematically,

 

( Function has to be analytic at all the interior points and on the contour C as well.)

To prove this, let's start with an analytic function f(z). Consider

 where C is any simple closed contour in the Z-plane. By knowing  
f(z) = u(x,y) + i v(x,y), and dz = dx + i dy , the above integral becomes 

 

where vectors V_1 = (u, -v) , V_2 = (v, u) and dr = (dx, dy).
Now, the two closed loop integrals on the right hand side can be converted into surface integrals, by using Stokes theorem. Hence, we can write,

 

 We can realize that these two integrands goes to zero, since the function  f(z), is an analytic function. Hence it satisfies the Cauchy-Riemann conditions,

So, for an analytic function, it is guaranteed that,

 

which implies

 

Hence the theorem is proved.